The Osgood condition for stochastic partial differential equations

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Foondun, Mohammud and Nualart, Eulalia (2021) The Osgood condition for stochastic partial differential equations. Bernoulli - Journal of the Bernoulli Society. pp. 295-311. ISSN 1350-7265 (https://doi.org/10.3150/20-BEJ1240)

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Abstract

We study the following equation  (∂u(t,x))/∂t  = ∆u(t,x) + b(u(t,x)) + σW (t,x),t > 0 where σ is a positive constant and W is a space-time white noise. The initial condition u(0, x) = u0(x) is assumed to be a nonnegative and continuous function. We first study the problem on [0,\,1] with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman, our first result shows that the solution blows up in finite time if and only if  which is the well-known Osgood condition. We also consider the same equation on thewhole line and show that the above condition is sufficient for the nonexistence of globalsolutions. Various other extensions are provided; we look at equations with fractionalLaplacian and spatial colored noise in Rd.

  • Item type:Article
    ID code:69333
    Dates:
    Date
    Event
    1 February 2021
    Published
    28 July 2019
    Accepted
    29 July 2018
    Submitted
    Subjects:Science > Mathematics
    Department:Faculty of Science > Mathematics and Statistics
    Depositing user: Pure Administrator
    Date deposited:13 Aug 2019 10:51
    Last modified:18 Dec 2024 02:30
    URI:https://strathprints.strath.ac.uk/id/eprint/69333
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